CSE 421
Algorithms
Richard Anderson
Lecture 8
Greedy Algorithms: Homework
Scheduling and Optimal Caching
Greedy Algorithms
• Solve problems with the simplest possible
algorithm
• The hard part: showing that something
simple actually works
• Today’s problems (Sections 4.2, 4.3)
– Homework Scheduling
– Optimal Caching
Homework Scheduling
• Tasks to perform
• Deadlines on the tasks
• Freedom to schedule tasks in any order
• Can I get all my work turned in on time?
• If I can’t get everything, I want to minimize
the maximum lateness
Scheduling tasks
•
•
•
•
Each task has a length ti and a deadline di
All tasks are available at the start
One task may be worked on at a time
All tasks must be completed
• Goal: minimize maximum lateness
– Lateness = fi – di if fi >= di
Example
Time
Deadline
2
2
4
3
2
Lateness 1
3
3
2
Lateness 3
Determine the minimum lateness
Time
Deadline
6
2
4
3
4
5
5
12
Greedy Algorithm
• Earliest deadline first
• Order jobs by deadline
• This algorithm is optimal
Analysis
• Suppose the jobs are ordered by deadlines,
d1 <= d2 <= . . . <= dn
• A schedule has an inversion if job j is scheduled
before i where j > i
• The schedule A computed by the greedy
algorithm has no inversions.
• Let O be the optimal schedule, we want to show
that A has the same maximum lateness as O
List the inversions
Time
a1
3
a2
a3
a4
Deadline
4
4
5
2
6
12
5
a4
a2
a1
a3
Lemma: There is an optimal
schedule with no idle time
a4
a2
a3
a1
• It doesn’t hurt to start your homework early!
• Note on proof techniques
– This type of can be important for keeping proofs clean
– It allows us to make a simplifying assumption for the
remainder of the proof
Lemma
• If there is an inversion i, j, there is a pair of
adjacent jobs i’, j’ which form an inversion
Interchange argument
• Suppose there is a pair of jobs i and j, with
di <= dj, and j scheduled immediately
before i. Interchanging i and j does not
increase the maximum lateness.
j
di dj
i
i
di dj
j
Proof by Bubble Sort
d1
a1
d2
d3
d4
a2
a4
a2
a4
a2
a1
a2
a1
a2
Determine maximum lateness
a3
a1
a1
a3
a4
a3
a3
a4
a3
a4
Real Proof
• There is an optimal schedule with no
inversions and no idle time.
• Let O be an optimal schedule k inversions,
we construct a new optimal schedule with
k-1 inversions
• Repeat until we have an optimal schedule
with 0 inversions
• This is the solution found by the earliest
deadline first algorithm
Result
• Earliest Deadline First algorithm
constructs a schedule that minimizes the
maximum lateness
Homework Scheduling
• How is the model unrealistic?
Extensions
• What if the objective is to minimize the
sum of the lateness?
– EDF does not seem to work
• If the tasks have release times and
deadlines, and are non-preemptable, the
problem is NP-complete
• What about the case with release times
and deadlines where tasks are
preemptable?
Optimal Caching
• Caching problem:
– Maintain collection of items in local memory
– Minimize number of items fetched
Caching example
A, B, C, D, A, E, B, A, D, A, C, B, D, A
Optimal Caching
• If you know the sequence of requests,
what is the optimal replacement pattern?
• Note – it is rare to know what the requests
are in advance – but we still might want to
do this:
– Some specific applications, the sequence is
known
– Competitive analysis, compare performance
on an online algorithm with an optimal offline
algorithm
Farthest in the future algorithm
• Discard element used farthest in the future
A, B, C, A, C, D, C, B, C, A, D
Correctness Proof
• Sketch
• Start with Optimal Solution O
• Convert to Farthest in the Future Solution
F-F
• Look at the first place where they differ
• Convert O to evict F-F element
– There are some technicalities here to ensure
the caches have the same configuration . . .
Subsequence Testing
• Is a1a2…am a subsequence of b1b2…bn ?
– e.g. S,A,G,E is a subsequence of
S,T,U,A,R,T,R,E,G,E,S
Greedy Algorithm for
Subsequence Testing